10 September 2001 Physics Letters A 288 (2001) 1–3 www.elsevier.com/locate/pla Chaos in the quantum double well oscillator: the Ehrenfest view revisited Analabha Roy a , J.K. Bhattacharjee b,∗ a Department of Physics, Indian Institute of Technology, Kanpur 208 016, India b Department of Theoretical Physics, Indian Association for the Cultivation of Science, Kolkata 700 032, India Received 3 April 2001; accepted 22 May 2001 Communicated by P.R. Holland Abstract We treat the double well quantum oscillator from the standpoint of the Ehrenfest equation but in a manner different from Pattanayak and Schieve. We show that for short times there can be chaotic motion due to quantum fluctuations, but over sufficiently long time there will be quantum noise induced oscillations between the two wells, thus providing an alternative approach to the quantum noise induced chaotic oscillations found recently by Bag and Ray.  2001 Elsevier Science B.V. All rights reserved. It is generally agreed that the foil quantum dynam- ics does not exhibit chaos. For systems which exhibit chaotic dynamics in the classical limit, it was clearly established by Fishman, Grempel and Prange [1] that there exists a critical time t ∼ 0 (1/h 1/r ) beyond which the dynamics crosses over to the quantal be- haviour. The exponent r was found to be 6.039 for systems which showed period-doubling bifurcations in the classical limit and 3.04 for the disappearance of the final KAM trajectory in the standard map. It was con- jectured in the late eighties that there could be systems where the classical dynamics is obviously regular but the semiquantal dynamics can be chaotic (for a more precise explanation of the term “semiquantal”, cf. Pat- tanayak and Schieve [2]). In support of this conjecture, Pattanayak and Schieve [2] explored the semiquantal dynamics of the double well oscillator governed by the * Corresponding author. Hamiltonian H = P 2 2 - 1 2 x 2 + λ 4 x 4 . The classical dynamics of this oscillator is obvi- ously regular, as explained in Landau–Lifshitz (Vol. 1), being a periodic trajectory centered about x =±1/ √ λ for total energy E in the range 0 >E> -1/4 and about x = 0 for E> 0. It was shown, based on an Ehrenfest equation approach that in the semiquantal limit the quantum fluctuations cause the dynamics of this oscillator to become chaotic with four repelling zones in the phase space. Now, the full quantum dy- namics of this oscillator should be regular just as all other fully quantum dynamics. Hence, we believe that the dynamics of this oscillator should cross over from a chaotic dynamics to a regular dynamics as one goes from a semiquantal to a fully quantum limit. The crossover should be characterized by a time t 0 . Unlike the cases treated by Grempel et al. [1], we believe that the time scale here should be exponentially big. By us- 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0375-9601(01)00351-6